A182740 A shell model of partitions as a table of partitions.

1, 2, 1, 3, 0, 1, 2, 0, 1, 1, 4, 0, 0, 1, 1, 3, 0, 2, 1, 1, 1, 5, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 1, 4, 0, 0, 2, 1, 1, 1, 1, 1, 3, 0, 2, 0, 0, 1, 1, 1, 1, 1, 6, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 0, 0, 0, 2, 1, 1, 1, 1, 1, 1, 1, 5, 0, 0, 3, 2, 0, 1, 1, 1, 1, 1, 1, 1

Offset

1,2

Comments

This array read by antidiagonals is a table of partitions for all integers.

For another version but as a binary code see A182741.

For more information see A135010 and A138121 which are the main entries for this sequence.

Links

Table of n, a(n) for n=1..91.

Omar E. Pol, Illustration of the shell model of partitions for 1..6 (2D-3D view)

Omar E. Pol, Illustration of the shell model of partitions for 1..10 (2D view)

Omar E. Pol Illustration of the shell model of partitions for 1..9 (3D view)

Example

For the numbers 1..6 the shell model of partitions has 6 shells. The model as a table looks like this:
  1 1 1 1 1 1
  2 . 1 1 1 1
  3 . . 1 1 1
  2 . 2 . 1 1
  4 . . . 1 1
  3 . . 2 . 1
  5 . . . . 1
  2 . 2 . 2 .
  4 . . . 2 .
  3 . . 3 . .
  6 . . . . .
Then replace the dots by zeros.
Remarks: one number by column, for example 23 is located only in a column, not in two columns.
The table looks like this:
  1 1 1 1 1 1
  2 0 1 1 1 1
  3 0 0 1 1 1
  2 0 2 0 1 1
  4 0 0 0 1 1
  3 0 0 2 0 1
  5 0 0 0 0 1
  2 0 2 0 2 0
  4 0 0 0 2 0
  3 0 0 3 0 0
  6 0 0 0 0 0
Array begins:
  1,1,1,1,1,1,
  2,0,1,1,1,
  3,0,0,1,
  2,0,2,
  4,0,
  3,

Prog

(PARI) a(max_part)={
  my(p=vecsort(partitions(max_part)), m=matrix(#p, max_part));
  for(i=1, #p, my(j=1);
    forstep(k=#p[i], 1, -1, m[i, j] = p[i][k]; j += p[i][k]));
  for(i=1, #p, print(m[i, ]));
  concat(vector(#p, i, vector(i, j, if(j <= max_part, m[i+1-j, j], 1))))
}; /* Martin Fuller, Jan 27 2026 */

Crossrefs

Cf. A135010, A138121, A182741, A182742, A182743.

Sequence in context: A317367 A071431 A277322 * A228786 A140699 A140256

Adjacent sequences: A182737 A182738 A182739 * A182741 A182742 A182743

Keyword

nonn,tabl

Author

Omar E. Pol, Nov 30 2010

Extensions

a(16) onward from Martin Fuller, Jan 27 2026

Status

approved